University of Petra Faculty of Information Technology

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1 University of Petra Faculty of Information Technology Measuring Algorithm Efficiency Dr. techn. Dipl. Inform. Bassam Haddad d Associate Professor of Computer Science Faculty of Information Technology Petra University Office: 731 Ext. 340 hddd@ haddad@uop.edu.jod Algorithm- Analysis DS-V Course No.: 6011 Prerequisite:6011 PL-II 010-I Based on: Data Structures and Abstractions with JAVA Frank M. Carrano & W. Savitech, Pearson Education, Ed, (1)+(1+1)+(1+1+1)+ Three algorithms for computing n for an integer n > 0. Sum(3)=3*(3+1)/ = 6 n: Size of the Problem Bassam Haddad 010 Bassam Haddad 010 Dr B. Haddad 1

2 Measuring Algorithm Efficiency Measuring Algorithm Efficiency The number of operations required by the algorithms The number of operations required by the algorithms As a programmers we are interested to the notice the effect of an inefficient Alg. not when the size of the Problem is small but when the problem is large. n +n+1 behaves like n when n is large because n is much larger than n+1 Bassam Haddad Bassam Haddad Dr B. Haddad

3 Big Oh Notation Big Oh Notation Note: till now, we cannot compute the actual time requirement of an Alg (why we have not jet implemented in JAVA but we found a Function) Such Function of the size of the problem, which behaves like the algorithm's actual time requirement If the time requirement increases, the value of the function increases. The value of the function is directly proportional to the time requirement Such function is called growth rate function: it measures how the time requirement grows as the problem size grow (asymptotic growth rate or Order) By comparing growth rate functions we can determine which Alg. is faster Estimation can be made for worst case, best case and average case Formal definition of Big Oh An algorithm's time requirement f(n) is of order at most g(n) f(n) = O(g(n)) For a positive real number c and positive integer N exist such that f(n) c g(n) for all n N An Algorithm A has a worst-case time requirement proportional to n We say A is O(n) Read "Big Oh of n" Bassam Haddad Bassam Haddad Dr B. Haddad 3

4 Big Oh Notation Big-Oh Rules Example: when n 3, f(n)=5n + 3 5n + n=6n N=3 f(n) c g(n) f(n) if O(n) Note: f(0) =3, f(1)= 8, f()=18,f(4)=3 6.g(0)=0, 0 6.g(1)=6, 6 6.g()=1, 6.g(3)=18, 6.g(4)=4 f(n) 6.g(n)= 6n for N=3 An illustration of the definition of Big Oh when n 1, f(n)=5n + 3 5n + 3n=8n N=1 f(n) if O(n) c g(n) Bassam Haddad Bassam Haddad Dr B. Haddad 4

5 Some Rules The following identities hold for Big Oh notation: O(k f(n)) = O(f(n)) O(f(n)) + O(g(n)) = O(f(n) + g(n)) O(f(n)) O(g(n)) = O(f(n) g(n)) Comments on Efficiency A programmer can use O(n ), O(n 3 ) or O( n ) as long as the problem size is small For Example: At the rate of one million operations per second: AnO(n ) algorithm would take one second to solve a problem size of 1000 An O(n 3 )algorithmwouldtakeone take one second to solve a problem size of 100 An O( n ) algorithm would take one second to solve a problem size of 0 Bassam Haddad Bassam Haddad Dr B. Haddad 5

6 Complexities of Program constructs Complexities of Program constructs The time complexity of a sequence of statements in a program is the sum of the statements complexities. However, it is sufficient to take the max of these complexities O(max(f 1,f,.,f n )) if fi the growth-rate function for a sequence statements: if (condition) S 1 else S The time complexity of such a program construct is the sum of the complexity of the condition and S 1 or S Max(f condition, f S1,f S ) S 1,S,,S n Bassam Haddad Bassam Haddad Dr B. Haddad 6

7 Complexities of Program constructs Other Notations The time complexity of a loop is the complexity of it body times the number of times the body executes Big-Oh: The maximum time requirement for an Algorithm O(g(n)) for i = 1 to n S O( nf S S( (n) ) =O(f S (n) ) Big Omega: For producing the minimum time requirement for an Algorithm Ω(g(n)) Big Theta: Big Theta analysis estimates the average case time Θ(g(n)) Bassam Haddad Bassam Haddad Dr B. Haddad 7

8 Big Oh Notation Big-Oh Rules If is f(n) a polynomial of degree k, then f(n) is O(n k ), i.e., 1. Drop lower-order terms.drop constant factors Use the smallest possible class of functions Say nis O(n) instead of n is O(n ) Use the simplest expression of the class Say 3n + 5 is O(n) instead of 3n + 5 is O(3n) Typical growth-rate functions evaluated at increasing values of n, Logarithms given base Bassam Haddad Bassam Haddad Dr B. Haddad 8

9 Big-Oh Rules The Efficiency of the implementation of the List Example: when n 3, f(n)=5n + 3 5n + n=6n N=3 f(n) if O(n) f(n) c g(n) when n 1, f(n)=5n + 3 5n + 3n=8n public boolean add(object newentry) { boolean success = true; // success is true at the beginning if (!isfull()) // The list is not full then { entry[length]= th] newentry; // assigns newentry to the current index // at the Beg. length =0 length++; //increase the counter } else success =false; return success; } N=1 c g(n) f(n) if O(n) Bassam Haddad Bassam Haddad Dr B. Haddad 9

10 The Efficiency of the implementation of the List The Efficiency of the implementation of the List public boolean add(int newposition, Object newentry) { boolean success=true; //we have to make space and shift. // The entires to higher pos. starting //with last entry position until to newposition if (!isfull() && (newposition >=1) && (newposition <= length+1) ) } {makespace(newposition); entry[newposition-1] = newentry; length++;} else success =false; return success; public boolean add(object newentry) // add at the End. we use the Private //Method getnodeat(length) { Node newnode= new Node(newEntry); if(isempty()) ()) firstnode= newnode; else { Node lastnode=getnodeat(length); //if not empty } lastnode.next=newnode; } length++; return true; O(n) O(n) Bassam Haddad Dr B. Haddad 10

11 Efficiency of Implementations of ADT List Comparing Implementations For array-based implementation Add to end of list Add to list at given position O(n) For linked implementation Add to end of list O(n) Add to list at given position O(n) Retrieving an entry O(n) The time efficiencies of the ADT list operations for two implementations, expressed in Big Oh notation Bassam Haddad Bassam Haddad 010 Dr B. Haddad 11

12 n n n A visual justification of n i = 1 n( n + 1) i = The sum represents the area covered by the marked Triangle n 1 n + n n(n+1) + n( ) = = A German elementary schoolteacher asked the 9-10 years old pupils to add up 1 to 100 But the answer cam immediately from Karl Gauss as a student based on the introduced formula (= 5,050 ) Bassam Haddad Dr B. Haddad 1